Under what conditions is the following flow field incompress

Under what conditions is the following flow field incompressible: V = (u, v, w) = (ax^2y)i + cxy^2j + dx^2yk

Solution

We have u = ax²y, v = cxy², w = dx²y

For incompressibility, delu/delx + delv/dely + delw/delz = 0

delu/delx = 2axy

delv/dely = 2cxy

delw/delz = 2dxy

We get 2axy + 2cxy + 2dxy = 0

or 2xy(a+c+d) = 0

Thus, the flow field will be incompressible when a+c+d = 0